Q:

In the figure shown, square WXYZ is inscribed in circle o. Also OM is perpendicular to XY and OM = 5. Find the area of the shaded region

Accepted Solution

A:
Answer:The area of the shaded region is 57.08 units squareStep-by-step explanation:- Lets explain some facts in the circle and the square- If a square inscribed in a circle, then the center of the circle is  the center of the square and the length of the diameter  of the circle equal the length of the diagonal of the square - The center of the square is the point of the intersection of its  diagonal- The length of the diagonal of a square = √2 the length of the side   of the square ⇒ d = √2 s* Lets solve the problem∵ WXYZ is a square inscribed in the circle O∴ The diameter of circle O = the diagonal of the square∵ WX is a side of the square WXYZ∵ XZ is a diagonal of square WXYS∴ XZ = √2 WX∵ OM ⊥ XY∵ O is the center of the circle and the square∴ OM = 1/2 the length of the side of the square∵ OM = 5∴ The length of the side of the square = 2 × 5 = 10∴ WX = 10 units∴ XZ = √2 × 10 = 10√2 units∵ XZ is a diameter of circle O∴ The diameter of the circle = 10√2∵ The radius of the circle = 1/2 diameter∴ The radius of the circle = 1/2 × 10√2 = 5√2 units∵ Area of the circle = πr²∴ The area of the circle = π (5√2 )² = 50π units²∵ The length of the side of the square is 10 units∵ The area of the square = s²∴ Area of the square = 10² = 100 units²∵ Area the shaded = area circle - area square∴ Area the shaded = 50π - 100 = 57.079 units²* The area of the shaded region is 57.08 units square